It just seems to me that Frodo reading that book could theoretically occur anytime between Bilbo adopting him and the Grey Havens, although clearly Frodo is too young for Wood to play him at the start of that time.

A mechanism that could be used, were the timeframe for the reading to be set, say 15 years before FOTR, might allow us to see other events we know happened, and the brief introduction of other characters or story fills. That sort of thing really would not surprise me. Mawr yw ein braint i berthyn i'r gwm Llynfi